You probably can get in on the VWS, as others have said. Many UK posters seem to treat arrests, convictions, jailings, as rather a joke; Americans do not.
The fact your BF has been jailed twice, and with 100% certainty, that means he is not eligible to enter legally on the VWS. He almost certainly would not get a visa if he went to the embassy either.
Consensus would say, go in on the VWS, they won't know, don't worry. I'd say, they're probably right, but if USC&BP do, expect to be deported and never allowed back.